About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 768272, 10 pages
http://dx.doi.org/10.1155/2012/768272
Research Article

Iterative Algorithms for General Multivalued Variational Inequalities

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China
2Mathematics Department, College of Science, King Saud University, Riyadh 1145, Saudi Arabia
3Mathematics Department, COMSATS Institute of Information Technology, Islamabad 44000, Pakistan
4Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
5Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 21 October 2011; Accepted 1 November 2011

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Yonghong Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce and study some new classes of variational inequalities and the Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational in equalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed.

1. Introduction

Variational inequality problems were initially studied by Stampacchia in 1964. Variational inequalities have applications in diverse disciplines such as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance, see [133] and the references therein. Variational inequalities have been extended and generalized in several directions using novel and innovative techniques. It is a common practice to study these variational inequalities in the setting of convexity. It has been observed that the optimality conditions of the differentiable convex functions can be characterized by the variational inequalities. In recent years, it has been shown that the minimum of the differentiable nonconvex functions can also be characterized by the variational inequalities. Motivated and inspired by these developments, Noor [19] has introduced a new type of variational inequality involving two nonlinear operators, which is called the general variational inequality. It is worth mentioning that this general variational inequality is remarkable different from the so-called general variational inequality which was introduced by Noor [16] in 1988. Noor [19] proved that the general variational inequalities are equivalent to nonlinear projection equations and the Wiener-Hopf equations by using the projection technique. Using this equivalent formulation, Noor [19] suggested and analyzed some iterative algorithms for solving the special general variational inequalities and further proved these algorithms have strong convergence. In this paper, we introduce and consider a new class of variational inequalities, which is called the general multivalued variational inequality. Using essentially the projection technique, we establish the equivalence between the multivalued variational inequalities and the multivalued Wiener-Hopf equations.

Related to the variational inequalities, we have the problem of finding the fixed points of the nonexpansive mappings, which is the subject of current interest in functional analysis. It is natural to consider a unified approach to these two different problems. Noor and Huang [21] considered the problem of finding the common element of the set of the solutions of variational inequalities and the set of the fixed points of the nonexpansive mappings. We use the Wiener-Hopf technique to suggest and analyze some iterative methods for finding the common element the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the special general variational inequalities. We also consider the convergence criteria of the proposed algorithms under suitable conditions. Several special cases are also discussed.

2. Preliminaries

Let be a nonempty closed convex subset of a real Hilbert space . Let be a multivalued mapping. Let be two nonlinear operators. We consider the problem of finding and such that Inequality of type (2.1) is called the general multivalued variational inequality. We will denote the set of solutions of the special general variational inequality (2.1) by . The general multivalued variational inequality (2.1) can be written in the following equivalent form, that is, find and such that This equivalent formulation is very important and plays a crucial role in the development of the iterative methods for solving the general multivalued variational inequalities.

We now discuss several special cases.

Special Cases
(A) If , then (2.1) reduces to: find such that which is called the general variational inequality, introduced and studied by Noor [19]. It has been shown that the minimum of a class of differentiable functions can be characterized by the general variational inequality of type (2.3).
(B) If , the identity operator, then (2.1) reduces to find and such that which is known as the mildly nonlinear multivalued variational inequality and has been studied extensively.
If and are single-valued nonlinear operators, then problem (2.1) is equivalent to finding such that which is known as the mildly nonlinear variational inequality, the origin of which can be traced back to Noor [15].
(C) If and , then (2.1) reduces to: find such that which is wellknown as the variational inequality, originally introduced and studied by Stampacchia [24] in 1964. It is clear from the above discussion that general multivalued variational inequality is quite general one. It has been shown that a wide class of problems arising in various discipline of mathematical and engineering sciences can be studied via the general multivalued variational inequalities (2.1) and its special cases.

In the sequel, we need the following well-known lemma.

Lemma 2.1. For a given satisfies the inequality if and only if where is the projection of into the closed convex set .

By using Lemma 2.1, one can prove that the general multivalued variational inequality (2.1) is equivalent to the following fixed point problem.

Lemma 2.2. is a solution of the special general variational inequality (2.1) if and only if satisfies the relation where is a constant.

Related to the general multivalued variational inequality (2.1), we consider the problem of solving the Wiener-Hopf equations. Let be two nonlinear operators and be a multi-valued relaxed monotone operator. Let , where is the identity operator. We consider the problem of finding such that which is called the special general multivalued Wiener-Hopf equations. We use to denote the set of solutions of the special general multivalued Wiener-Hopf equations. For different and suitable choice of the operators , we can obtain various forms of the Wiener-Hopf equations, which have been studied by Noor [17], Shi [22], and others.

Using essentially the technique of Noor [17, 18] and applying Lemma 2.2, one can establish the equivalence between the Wiener-Hopf equations and the general multivalued variational inequalities (2.1). To convey an idea of the technique and for the sake of completeness, we include its proof.

Lemma 2.3. If , then and satisfy the general Wiener-Hopf equations (2.10), where where is a constant.

Proof. Let . Then, from Lemma 2.2, we have Let Then, we have Therefore, from (2.13), we obtain It follows that where , which is exactly the general Wiener-Hopf equations (2.10). This completes the proof.

Remark 2.4. Let be a nonexpansive mapping. If , then one can easy to see which is implies that where is a sequence in .

Using Remark 2.4 and Lemma 2.3, we can suggest the following algorithm for finding the common element of the solutions set of the variational inequalities and the set of fixed points of a nonexpansive mapping.

Algorithm 2.5. For a given   arbitrarily, let the sequence be generated by where is a sequence in and is some constant.

Note that, if , then Algorithm 2.5 reduces to the following iterative method for solving the general variational inequalities.

Algorithm 2.6. For a given   arbitrarily, let the sequence be generated by where is a sequence in and is some constant.

If , then Algorithm 2.5 reduces to the following iterative method for solving the general variational inequalities (2.3), which was considered by Noor [19].

Algorithm 2.7. For a given   arbitrarily, let the sequence be generated by where is a sequence in and is some constant.

If and , then Algorithm 2.5 reduces to the following iterative method for solving the variational inequalities (2.6).

Algorithm 2.8. For a given   arbitrarily, let the sequence be generated by where is a sequence in and is some constant.

We recall the well-known concepts. The multivalued mapping is said to be -Lipschitzian if there exists a constant such that Recall that a mapping is called nonexpansive if We will use to denote the set of fixed points of .

A mapping is called -strongly monotone if there exists a constant such that and -Lipschitz continuous if there exists a constant such that

3. Main Results

Now we state and prove our main result.

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let be an -strongly monotone and -Lipschitz continuous mapping, an -strongly monotone and -Lipschitz continuous mapping and be a -Lipschitz continuous mapping. Let be a nonexpansive mapping such that . Assume that where then the approximate solution obtained from Algorithm 2.5 converges strongly to .

Proof. Let . Then, from Remark 2.4, we have where and satisfy the general Wiener-Hopf equations (2.10).
From (2.19) and (3.1), we have From (2.19), we have Since is an -strongly monotone and -Lipschitz continuous mapping, we have where .
At the same time, we note that is an -strongly monotone and -Lipschitz continuous mapping, so we have From (3.5)–(3.7), we have where Using (3.1), we see that . Substituting (3.4) into (3.8), we have Since diverges and , we have . Consequently, the sequence converges strongly to in , the required result.

4. Conclusion

One of the most difficult and important problems in variational inequalities is the development of an efficient numerical methods. One of the technique is called the projection method and its variant forms. Projection method represent an important tool for finding the approximate solution of various types of variational inequalities. The projection type methods were developed in 1970s. The main idea in this techniques is to establish the equivalence between the variational inequalities and the fixed point problem using the concept of projection. These methods have been extended and modified in various ways. Shi [22] considered the problem of solving a system of nonlinear projections, which are called the Wiener-Hopf equations. It has been shown by Shi [22] that the Wiener-Hopf equations are equivalent to the variational inequalities. It turns out that this alternative formulation is more general and flexible. It has been shown that the Wiener-Hopf equations provide us a simple, natural, elegant, and convenient device to develop some efficient numerical methods for solving variational and complementarity problems. In this paper, we introduce and study some new classes of variational inequalities and Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational inequalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed. The ideas and techniques of this paper may be a starting point for a wide range of novel and innovative applications in various fields.

Acknowledgments

The authors thank the referees for useful comments and suggestions. The research of Professor M. Aslam Noor is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia, and Research Grant: KSU.VPP.108.

References

  1. L. C. Ceng, Q. H. Ansari, and J. C. Yao, “Relaxed extragradient iterative methods for variational inequalities,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 1112–1123, 2011. View at Publisher · View at Google Scholar
  2. L. C. Ceng, M. Teboulle, and J. C. Yao, “Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems,” Journal of Optimization Theory and Applications, vol. 146, no. 1, pp. 19–31, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. S. Chang, H. W. J. Lee, C. K. Chan, and J. A. Liu, “A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6830–6837, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. F. Cianciaruso, G. Marino, L. Muglia, and Y. Yao, “On a two-step algorithm for hierarchical fixed point problems and variational inequalities,” Journal of Inequalities and Applications, vol. 2009, Article ID 208692, 13 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. 1-2 of Springer Series in Operations Research, Springer, New York, NY, USA, 2003.
  6. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, 1984.
  7. B. S. He, “A new method for a class of linear variational inequalities,” Mathematical Programming, vol. 66, no. 2, pp. 137–144, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. A. N. Iusem and B. F. Svaiter, “A variant of Korpelevich's method for variational inequalities with a new search strategy,” Optimization, vol. 42, no. 4, pp. 309–321, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. P. Jaillet, D. Lamberton, and B. Lapeyre, “Variational inequalities and the pricing of American options,” Acta Applicandae Mathematicae, vol. 21, no. 3, pp. 263–289, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. E. N. Khobotov, “Modification of the extra-gradient method for solving variational inequalities and certain optimization problems,” USSR Computational Mathematics and Mathematical Physics, vol. 27, no. 5, pp. 120–127, 1987.
  11. G. M. Korpelevich, “An extragradient method for finding saddle points and for other problems,” Ekonomika i Matematicheskie Metody, vol. 12, no. 4, pp. 747–756, 1976.
  12. P. Kumam, N. Petrot, and R. Wangkeeree, “Existence and iterative approximation of solutions of generalized mixed quasi-variational-like inequality problem in Banach spaces,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7496–7503, 2011. View at Publisher · View at Google Scholar
  13. X. Lu, H. K. Xu, and X. Yin, “Hybrid methods for a class of monotone variational inequalities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1032–1041, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. G. Marino, L. Muglia, and Y. Yao, “Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points,” Nonlinear Analysis, Theory, Methods and Applications. In press. View at Publisher · View at Google Scholar
  15. M. A. Noor, On Variational Inequalities, Brunel University, London, UK, 1975.
  16. M. A. Noor, “General variational inequalities,” Applied Mathematics Letters, vol. 1, no. 2, pp. 119–122, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. A. Noor, “Wiener-Hopf equations and variational inequalities,” Journal of Optimization Theory and Applications, vol. 79, no. 1, pp. 197–206, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. M. A. Noor, “Some developments in general variational inequalities,” Applied Mathematics and Computation, vol. 152, no. 1, pp. 199–277, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. M. A. Noor, “Differentiable non-convex functions and general variational inequalities,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 623–630, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. M. A. Noor and E. A. Al-Said, “Wiener-Hopf equations technique for quasimonotone variational inequalities,” Journal of Optimization Theory and Applications, vol. 103, no. 3, pp. 705–714, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. M. A. Noor and Z. Huang, “Wiener-Hopf equation technique for variational inequalities and nonexpansive mappings,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 504–510, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  22. P. Shi, “Equivalence of variational inequalities with Wiener-Hopf equations,” Proceedings of the American Mathematical Society, vol. 111, no. 2, pp. 339–346, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. V. Solodov and B. F. Svaiter, “A new projection method for variational inequality problems,” SIAM Journal on Control and Optimization, vol. 37, no. 3, pp. 765–776, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. G. Stampacchia, “Formes bilineaires coercitives sur les ensembles convexes,” Comptes Rendus de l'Academie des Sciences, vol. 258, pp. 4413–4416, 1964. View at Zentralblatt MATH
  25. R. U. Verma, “Projection methods, algorithms, and a new system of nonlinear variational inequalities,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 1025–1031, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. H. K. Xu and T. H. Kim, “Convergence of hybrid steepest-descent methods for variational inequalities,” Journal of Optimization Theory and Applications, vol. 119, no. 1, pp. 185–201, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  27. J. C. Yao, “Variational inequalities with generalized monotone operators,” Mathematics of Operations Research, vol. 19, no. 3, pp. 691–705, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  28. Y. Yao, R. Chen, and H. K. Xu, “Schemes for finding minimum-norm solutions of variational inequalities,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 7-8, pp. 3447–3456, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  29. Y. Yao, Y. C. Liou, and S. M. Kang, “Two-step projection methods for a system of variational inequality problems in Banach spaces,” Journal of Global Optimization. In press. View at Publisher · View at Google Scholar
  30. Y. Yao, M. A. Noor, and Y. C. Liou, “Strong convergence of a modified extra-gradient method to the 4 minimum-norm solution of variational inequalities,” Abstract and Applied Analysis, vol. 2012, Article ID 817436, 9 pages, 2012. View at Publisher · View at Google Scholar
  31. Y. Yao, M. A. Noor, K. I. Noor, Y. C. Liou, and H. Yaqoob, “Modified extragradient methods for a system of variational inequalities in Banach spaces,” Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1211–1224, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  32. Y. Yao and N. Shahzad, “New methods with perturbations for non-expansive mappings in Hilbert 5 spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 79, 2011.
  33. Y. Yao and N. Shahzad, “Strong convergence of a proximal point algorithm with general errors,” Optimization Letters. In press. View at Publisher · View at Google Scholar